Now let’s do some more interesting examples of geometric quantization using the functor described in Part 4. Let’s look at the spin-j particle with j > 1/2.
To be specific, let’s consider the spin-3/2 particle. There’s nothing special about the number 3 here: everything I’ll say can be generalized. But the number 3 will give me a nice excuse to show you a picture of a curve called the ‘twisted cubic’.
We can build a spin-3/2 particle from three spin-1/2 particles, all having angular momenta pointing in the same direction. Classically this procedure amounts to tripling a vector, but quantum-mechanically it’s related to cubing. This is a bit mysterious to me, but let me explain.
Classically, if we take a vector of length 1/2 in and triple it we get a vector of length 3/2. This gives a map from classical states of the spin-1/2 particle to classical states of the spin-3/2 particle. In other words: a map from the sphere of radius 1/2 to the sphere of radius 3/2. Simple! But this is not a symplectic map, since as we saw last time, we are giving the latter sphere a symplectic structure that’s 3 times as big. So, it’s not a valid classical process. Still, it’s a perfectly fine way to get our hands on lots of states of the classical spin-3/2 particle! All of them, in fact.
Quantum-mechanically the state space for the spin-1/2 particle is We can think of a guy in here as a linear functional on the dual We can cube this and get a homogeneous polynomial of degree 3 on The space of such polynomials is called and this is the space of states of the quantum spin-3/2 particle. So, we get a map
Simple! This describes the process of ‘triplicating’ the state of spin-1/2 particle and getting a spin-3/2 particle. But this is not a linear map, so it’s not a valid quantum process. As the saying goes, “you can’t clone a quantum”. Still, it’s a perfectly fine way to get our hands on lots of states of the quantum spin-3/2 particle! But not all of them, as we’ll soon see.
Geometric quantization should reconcile and combine classical and quantum mechanics. How can we do this here?
It’s pretty simple. We projectivize the map
Cubing sends any line through the origin in into a unique line through the origin in So, we get a map from to the projective space of And we define the image of this map to be the space of classical states of the spin-3/2 particle!
Let’s see what it looks like. I’m thinking of an element as a linear functional
If we cube it we get this homogeneous polynomial of degree 3:
If we take and as our basis of the homogeneous polynomials of degree 3, we can identify with So we get
If we projectivize this map we get a map between projective varieties, which I’ll call
The image of this map is the space of states of the classical spin 3/2-particle! It’s a copy of sitting inside projective 3-space. But because cubing a nonlinear map, this copy will be twisted: not a ‘line’ but a ‘curve’. People call it the twisted cubic.
Just for kicks, let’s take a closer look at it. Let’s use homogeneous coordinates and write for the point in corresponding to a nonzero vector Similarly, any point in can be written as a nonzero 4-tuple of complex numbers with a bracket around it. We get
so the twisted cubic is
This is hard to visualize, so we can work with a copy of that’s dense inside namely the set where The portion of the twisted cubic sitting in this set is
This is still hard to visualize, so we can restrict to the reals and think about the curve
This is also called the twisted cubic! People often rescale the axis to get rid of the number 3 here.
This version of the twisted cubic is still a bit hard to visualize, but it’s the intersection of two very nice surfaces: the surface and the surface So, the twisted cubic is the black curve:
in this nice picture uploaded to ResearchGate by Alexander M. Kasprzyk.
Okay, back to serious business! Let’s call the twisted cubic It’s exactly the sort of thing we can geometrically quantize using our functor
Why? Remember, is defined to be the smallest linear subspace such that But twists around so much that the smallest that works is all of
Now let’s take stock of where we are and draw some general conclusions from what we’ve seen in this example. We now have a firm grip on the space of quantum states of the spin-3/2 particle:
and also the space of classical states of the spin-3/2 particle, the twisted cubic:
Now for something cool: the latter sits inside the projectivization of the former! This is obvious, but it has a very nice physical meaning. While I’ve been calling the space of quantum states of the spin-3/2 particle, it is very reasonable to argue that quantum states are actually points of its projectivization, I will skip the argument, which is old, famous and convincing:
• Wikipedia, Projective Hilbert space.
What matters for us here is that classical states of the spin-3/2 particle give some of these quantum states—far from all, but some of the nicest ones! They are the states obtained by ‘cubing’ a state of a spin-1/2 particle. In other words, they are the states where we can think of our spin-3/2 particle as made of three spin-1/2 particles with their spins perfectly aligned.
It’s nice to say this with a bit more physics jargon. These quantum states coming from classical ones are called ‘coherent states’ . A general quantum state of the spin-3/2 particle is a ‘quantum superposition’ of these coherent states. By this, I mean that the smallest linear subspace for which contains the twisted cubic is all of
And while we’ve seen this in a particular example, it’s a completely general feature of our setup! Remember, projectivization
is the left adjoint of quantization
This means that for any we have
We get something interesting if we take here. Since we get
Category theorists call this inclusion of in the ‘unit’ of our pair of adjoint functors. It says how classical states sit inside the projectivization of If we call points of quantum states, those in are called the coherent states.
Moreover every quantum state is a ‘quantum superposition’ of coherent states! In other words, the smallest linear subspace for which sits inside is all of
Why is this true? It’s just the definition of
So, I hope you see how much physics is packed into these adjoint functors and I’ll do more examples next time. In the meantime, you can read more about this approach to the spin-3/2 particle here:
• Dorje C. Brody and Lane P. Hughston, Geometric quantum mechanics.
I was greatly inspired by this article!
• Part 1: the mystery of geometric quantization: how a quantum state space is a special sort of classical state space.
• Part 2: the structures besides a mere symplectic manifold that are used in geometric quantization.
• Part 3: geometric quantization as a functor with a right adjoint, ‘projectivization’, making quantum state spaces into a reflective subcategory of classical ones.
• Part 4: making geometric quantization into a monoidal functor.
• Part 5: the simplest example of geometric quantization: the spin-1/2 particle.
• Part 6: quantizing the spin-3/2 particle using the twisted cubic; coherent states via the adjunction between quantization and projectivization.